Can we always solve this equation in the space of Hermitian structures on a complex vector bundle?

Question

Let $E,F$ and $G$ be three complex $C^{\infty}$-vector bundles of rank $r,s$ and $rs$.

(I am using the notation from Kobayashi - Differential geometry of complex vector bundles, VI §2)

Assume we have an isomorphism $f: E\otimes F \stackrel{\sim}{\longrightarrow} G$.

If $E$ and $G$ are equipped with Hermitian structures, that is we have $(E,h)$ and $(G,h'')$ where $h$ and $h''$ are positive definite Hermitian bilinear forms on $E$ resp. $G$.

Can we always find a Hermitian structure $h'\in Herm^{+}(F)$, depending on $h$ and $h''$, such that $f$ gets an isometry? That is: do we get an isometry:

$\tilde{f}: (E\otimes F,h\otimes h') \stackrel{\sim}{\longrightarrow} (G,h'')$?

Otherwise stated: can we solve the equation $h\otimes x=h''$ for $x\in Herm^{+}(F)$?

Edit: What if we only have the metric $h''$? Can we find $x,y$ with $x\otimes y\cong h''$?

There is the following situation:

We have a sheaf of $O_M$-algebras $R$ and two $R$-modules $S,T$ such that $R,S,T$ and $\mathcal{H}om_R(S,T)$ are locally free over $M$. Furthermore we have an isomorphism of sheaves $\mathcal{H}om_{O_M}(S,T)\cong \mathcal{H}om_{O_M}(R,O_M)\otimes_{O_M} \mathcal{H}om_R(S,T)$. Now we have metrics on the bundles associated to $S$ and $T$. Can we find metrics on the bundles associated to $\mathcal{H}om_{O_M}(R,O_M)$ $\mathcal{H}om_R(S,T)$ such that the isomorphism on the associated bundles gets an isometry? Can we do something better/easier in this special situation? Or is there no hope for anything in this situation?

Answer 1

If $E$ has rank $0$ or $1$, 'yes', otherwise, 'no'. Just do a dimension count. You'll find that you have (many) more unknowns than equations and, for general $h$ and $h''$, there will be no solution $x$.

The answer for your modified question is still 'no, in general when $r$ and $s$ are both greater than $1$.' This is purely a pointwise linear algebra question: Consider a tensor product of complex vector spaces $V$ and $W$ of dimensions $r$ and $s$, respectively, say $V\otimes W$. The space of Hermitian metrics on $V\otimes W$ has dimension $(rs)^2$, while the space of Hermitian metrics that are tensor products of metrics on $V$ and $W$ only has dimension $r^2+s^2-1$. Thus, when $r$ and $s$ are greater than $1$, most metrics on $V\otimes W$ are not tensor product metrics.